Is it true that in the 50's the average man could provide for his family by himself? (In contrast to now, where both man and woman seem to need to work to provide for the family)

For the purposes of my answer, let's limit this to the United States, where we have excellent longitudinally-comparable data since the 1950s about labor.

Women did work, they just weren't included in labor statistics (See Corrections)

I think this point maybe should come first, although I didn't originally include it. Women did a lot of work that wasn't captured as market labor. That's still true, and is just gradually changing. The rise of women working has more to do with the growing market labor opportunities, changes in societal views on female roles, and changes in labor productivity, than it does with "the average man being able to provide for his family." You could rephrase the original question as, "Is it true that in the 1950s, many married women did not work, because society did not provide opportunities for them to be economically productive outside the home?" Regardless, I'll still attempt to tackle changes outside of the domain of this bullet point, below, which could probably be phrased as "what are the changes in economic opportunity for white male earners in the United States over the past 60 years"

Employment Changes

From the Bureau of Labor Statistics, prime-age male labor participation (men between 25-54) peaked in 1954 at 98% employment and has gradually declined over the next 64 years to today, where we now sit at around 88%. In other words, in the 1950s it is fair to say there was a lot more labor participation for men than there is today— only ~~2 in 100 men that were seeking employment would not have found it, compared to 12 in 100 today~~ only 2 in 100 men were not seeking employment, compared to 12 in 100 today.* (See Corrections) This leads us to the first and most compelling point about why single-family incomes in America seemed to work so well: for basically any prime age male in America, in 1950, if you wanted work you could find it. Also, those jobs were desirable enough that people would take them, or needed to take them, likely because other economic alternatives were not available (weaker social safety net, no economic opportunity for their wives, etc.) (See Corrections)

Purchasing Power Changes

A secondary factor is the relative decline of labor's purchasing power since the 1950s to today. To understand this, we need to contextualize wealth in 1950s America. There have always been wealthy people in America, but post-war economics suddenly upended the math, as a variety of economic factors had wage growth for labor suddenly accelerating rapidly, especially compared to income returns on wealth. (Piketty) This trend would continue through the 1970s, giving a nearly 20-year period in which labor-based income households (most "typical" households) saw each year giving them more income, greater buying power, and a rising standard of living. By comparison, real median wage has stagnated in purchasing power from 1980–2010. (Current Population Report, Census) A natural side effect of this, along with household consumption continuing to rise, led to greater pressure on households to diversify into two-income.

Women Entering the Workforce

Of difficult-to-quantify effects, we also have women entering the workforce over this time period, which in theory would increase total employable pools and put downward pressure on wages for labor. However, labor in America remains heavily gender-segregated, and it's less clear how impactful this is, although substantial amount of research goes into trying to describe this topic with greater specificity.

Sentimental and Inaccurate Media Depictions of Post-War America

Lastly, we should also be quick to point out that most of our beliefs of households in 1950 America are heavily skewed by somewhat rosy-retrospective depictions of white, suburban life in the 50s. Television and movies released in the 60s, 70s, and 80s did little to show the diverse, heterogenous, and challenging landscape of 1950s America, instead choosing to focus on the "white picket fence" America. In the same way that 21st century viewers of a show like "Friends" might mock a waitress in NYC renting a lavish Manhattan apartment, we should remind ourselves that "I Love Lucy" or "It's A Wonderful Life" had little to do with the realities of economic hardship faced by mid-20th century American families. Even complex portrayals of nostalgic America, like "Mad Men", don't exactly disabuse us of the reality that for many in America in 1950, economic conditions were challenging, quality of life was low (air conditioning, phone service, refrigeration, and television are all common lifestyles today that would have been unavailable to many or most households then), and jobs were often brutal and unsupportive of hardship. From "The Economic Progress of Black Men in America" we see the other side of the coin of the employment figures I referenced earlier: Black median household income was just $1900 in 1950, compared to $3400 for white households. This put greater pressure on wives to have earning power in poorer households.

Recap

To summarize: although it may be an exaggeration to say "the average 1950s man could provide for his family by himself", there were definitely large-scale macroeconomic factors that helped make that more true in 1950s America than today. Primarily in the form of near-guaranteed-employment, and higher and continual growth in buying power for labor, median single-family households probably "felt" better off in 1950 and 1960 than they do today. This has been further reinforced through media depictions, with heavy nostalgia and rosy-retrospection, of post-war white America.

Corrections

I'd like to thank /u/dmoni002, /u/jakfrist, and /u/gorbachev for their criticisms about this post, captured here. In particular, I'd like to emphasize: women entering the workforce was not a bad sign economically: it was a sign that there was greater economic opportunity for women, and less societal pressure to intentionally exclude women from the workplace. /u/dmoni002 put it better than I could:

Zero-sum lump of labor thinking; those women no longer have to beg their husbands to buy goods and services for them, those women are now directly buying things they want themselves with their own money. Again, if you have some village woman who no longer needs to spend 4 hours a day trekking for water because her home now has plumbing, and she spends those 4 hours working for an income – there’s both more production (S) and more spending (D). When the supply of labor exogenously increases, labor demand increases also so more women working will have an ambiguous effect on wages. Similarly, as we've already seen the total compensation throughout this period was rising! (Interesting thought: do women and men have different compensation preferences? Do women prefer a larger percentage of compensation to be paid as benefits?)

Answering macroeconomic history questions produces a lot of the same problems as answering controversial history questions, but we still shouldn't shy away from attempting— the AH format lends itself to the feeling of "certainty" with accepted answers, but anyone would tell you there are lots of nuances that you can't get into in a very brief reddit post. Still hope my answer helps both the OP and other readers on this topic.

Citations

White House Report on Long-Term Decline in Prime-Age Male Labor Force Participation, 2016
BLS Labor Force Participation Rate
Piketty, Thomas. Capital in the Twenty-First Century, 2013.
Current Population Report, US Census, see Real Income Trends
Economic Progress of Black Men in America, 1986